On Harmonic Maps Between Finsler Manifolds

Finsler geometry is just Riemannian geometry without quadratic restriction on the metrics ([12]), which was firstly introduced by B. Riemann in 1854. But he focused on the quadratic metrics which are called Riemannian metrics nowadays. In 1918, P. Finsler studied the curves and surfaces of general metrics in his doctor thesis ([17]) and then the name "Finsler geometry" was generally accepted. From 90's of the 20th century, Finsler geometry has been developed rapidly under the encouragement of S.-S Chern which generalized many important concepts and results in Riemannian geometry, such as the volume comparison theorems ([47]), harmonic maps ([33], [45], [20]), the geometry of submanifolds ([48], [21]), Ein-stein metrics ([2], [10]), Gauss-Bonnet theorems ([7]), pre-compactness theorems ([46]) and so on. Meanwhile, it has been applied to physics, biology, control theory, psychology and so on ([1], [3], [5], [9]).Harmonic maps are important both in differential geometry and mathemat-ical physics, which are the natural extensions of geodesics, minimal submanifolds and harmonic functions. Some open problems in Finsler geometry have been proposed in [32], one of which is to study harmonic maps between Finsler man-ifolds. Since then, there have been many works on it, such as the computations of the first and second variation formulas, the stability ([33], [45], [20], [44]), the existence ([34]), the regularity([38], [36]) and so on. Moreover, there are some works on harmonic maps between complex Finsler manifolds ([39], [19]).In this paper we study harmonic maps on real Finsler manifolds. The content is divided into four parts corresponding to four chapters. In the first chapter, we introduced the concept of focal points on Finsler manifolds which was ex-tended from Riemannian case and established a rigidity theorem between Finsler manifolds. In the second chapter, we study the regularity theories of energy min-imizing maps on Finsler manifolds. In the third chapter, we study the relations between harmonic maps and minimal submanifolds. In the last chapter, we study the properties of H-harmonic map. 0.1 Rigidity TheoremsFirst, we are going to introduce some definitions and main results of har-monic maps between Finsler manifolds. LetΦ:(M,F)→(M,F) be a non-degenerate smooth map between Finsler manifolds, i.e. ker(dΦ)= 0. By using the volume measure dVSM induced from the projective sphere bundle SM, we can define the energy of 0 as where Cn-1 denotes the volume of the unit Euclidean (n-1)-sphere Sn-1, dVsM=Ωdτ∧dx, LetΦt, t∈(-ε,ε) be a smooth variation ofΦwithΦ0=Φ,Φt|(?)M=Φ|(?)M with the variation field V =(?)|t=0.The first variation of the energy functional forΦcan be written as where Gk and Gαare the geodesic coefficients for (M,F) and (M,F), respectively. Harmonic maps are defined as the critical points of the energy functionals. HenceΦis harmonic if and only if for allβ. Particularly, ifτ(Φ)= 0,Φis called the strongly harmonic map.Φis called to be totally geodesic if f (Φ)= 0, i.e. b▽dΦ=0. If the target manifold is Riemannian, the assumption thatΦis non-degenerate is unnecessary. If both M and M are Riemannian, the harmonicity and strongly harmonicity are the same. The harmonic mapΦis called to be stable if its second variation of the energy functional is always nonnegative; otherwise unstable. ([45,20])The rigidity theorems for harmonic maps between Finsler manifolds were investigated firstly by an extrinsic average variational method. For example, it was proved that there is neither nonconstant stable harmonic map from a compact Finsler manifold to the superstrongly unstable manifold (including n(> 2)-dimansional Euclidean sphere) nor non-degenerate stable harmonic map from an n(>2)-dimansional Euclidean sphere to a Finsler manifold. In [22], the rigidity theorems were proved by the Bochner technique for non-degenerate map between Finsler manifolds. In [54], by establishing a convex function on Rie-mannian manifolds without focal points, the author provided another approach to get the rigidity theorems for harmonic maps between Riemannian manifolds. Riemannian manifolds with non-positive section curvatures have no focal points.We firstly introduced the concept of focal points on Finsler manifolds which was extended from Riemannian case ([40]). Let k(s),s∈(-∈,∈) and c(t), t∈[0,1] be geodesics on Finsler manifold (M, F), which are gc(0)-orthogonal at p = c(0). A vector field J(t) along c(t) is called a k-Jacobi field if it is the variation vector field of a variation of c(t) through geodesics which are initially perpendicular to k(s) with respect to gTs(0), where Ts(0) is a linearly parallel vector field along k(s) with To(0)= c(0). It is equivalent to saying that J(t) is a Jacobi field along c(t) satisfying the initial conditions gc(0)(DcJ(0),J(0))=0, gc(J(0),c(0))=0. A point c(to) is a focal point of k(s) along c(t) if there is a non-trivial k-Jacobi field along c(t) vanishing at c(to). If no such geodesic has a focal point on M, then M is said to have no focal points. Finsler manifolds without focal points have the following properties:Proposition 0.1 Let (M,F) be a forward complete Finsler manifold. (ⅰ) (M, F) has non-positive flag curvature if and only if (d2)/(dt2)gc(J, J)> 0 for every Jacobi field J(t) along every geodesic c(t);(ⅱ) If F is a reversible Finsler metric or a Berwald metric, then (M, F) has no focal points if and only if d/(dt)gc(J, J)> 0 for t> 0, where J(t) is a non-trivial Jacobi field along any geodesic c(t) vanishing at t.= 0;(ⅲ) (M, F) has no conjugate points if and only if gc(J, J)> 0 for t> 0, where J(t) is a non-trivial Jacobi field along any geodesic c(t) vanishing at t= 0.Moreover, Finsler manifolds with non-positive flag curvature must have no conjugate points. If (M, F) is a forward complete Finsler manifold which is equipped with either a reversible Finsler metric or a Berwald metric, then when (M, F) has non-positive flag curvature, it must have no focal points, when it has no focal points, it must have no conjugate points.A function f defined on an open set U of (M, F) is said to be convex ifτ(f)> 0 at each point on SU ([22]). When (M, F) is Riemannian, it is equivalent to saying that Vdf is a semi-positive definite form, which is the definition of convex function in Riemannian case. We will construct a convex function on Finsler manifolds without focal points:Proposition 0.2 Let (M, F) be a simply connected forward complete Finsler manifold without focal points which is equipped with either a reversible Finsler metric or a Berwald metric. Then the square of the distance function from any fixed point p∈M is convex on M\{p}.Remark 1. When (M,F) is Riemannian, the square of the distance function from any fixed point p∈M is convex on the whole manifold ([54]).Φ:M→M is a harmonic map between Riemannian manifolds if and only if it carries germs of convex functions on M to germs of subharmonic functions on M ([18]). We can get a similar result in Finsler case. Proposition 0.3 LetΦ:(M, F)→(M, F) be a map between Finsler manifolds, and f be a convex function on (M, F). (ⅰ) IfΦis a non-degenerate totally geodesic map, then∫SxMΔg(f oΦ)Ωdτ> 0; (ⅱ) If(M, F) is Riemannian andΦis strongly harmonic, thenΔg(foΦ)> 0, whereΔg denotes the Laplacian on (SM,g).By using Proposition 0.2 and Proposition 0.3 we can complete the proof of the main theorem of this chapter which is a generalization of the result in [54]. Theorem 0.4 Let (M, F) be a compact n-dimensional Finsler manifold with fi-nite fundamental group, and (M, F) be a forward complete m-dimensional Finsler manifold without focal points which is equipped with either a reversible Finsler metric or a Berwald metric. If n< m, then there is no non-degenerate totally geodesic map from (M, F) to (M, F). Moreover, if (M, F) is a complete Rieman-nian manifold without focal points, then any strongly harmonic map from(M, F) to (M,F) must be constant.0.2 Regularity TheoriesOne method to establishing the existence of harmonic maps between Rieman-nian manifolds is the direct method of the calculus of variations ([37], [23], [25]), and the regularity has been proved by C. B. Morrey ([37]), Schoen and Uhlen-beck ([42,43]). There are also many works on the existence and regularity of harmonic maps on Finsler manifolds. X. H. Mo and Y. Yang proved the fun-damental existence theorem for harmonic maps from Finsler manifolds to Rie-mannian manifolds in 2005 ([34]). Later, H. von der Mosel and S. Winklmann showed that weakly harmonic maps with image contained in a regular ball are locally Holder continuous ([38]). Recently, X. H. Mo and L. Zhao proved that the weakly harmonic map from a Finsler surface without boundary to a sphere is smooth actually ([36]). One approach to study the maps from Finsler manifolds to Riemannian manifolds is to construct a Riemannian metric from the funda-mental tensor of the Finsler metric, so these maps can be considered as the ones that from Riemannian manifolds with induced Riemannian metric and induced volume measure. More precisely, letΦ:(M, F)→(N, h) be a smooth map from an n-dimensional Finsler manifold (M, F) to an m-dimensional Riemannian manifold (N, h). The energy ofΦcan be written as Let (gij):=(gij)-1, then g:=gij(x)dxi (?) dxj is a Riemannian metric on M([34]).Hence where|dΦ|2/g denotes the norm with respect to g,i.e.|dΦ|2/g=gij(x)ΦiαΦjβhαβ(Φ(x)). Then the Finsler manifold(M,F)can be considered as a Riemannian manifold (M,g,σ)with induced Riemannian metric g and induced volume measureσ(x)dx. Harmonic maps are defined as the critical points of the energy functionals.By the Nash imbedding theorem we embed N isometrically in some Eu-clidean space Rk.After the direct computations,we will have several equivalent conditions for harmonic maps:Proposition 0.5 The following statements are equivalent: (ⅰ)the smooth mapΦis harmonic; (ⅱ)Φsatisfies the differential equationΔσΦα+gijTβγαΦiβΦjγ=0 for allα,whereΔσ=1/σ(?)/((?)xi))(σgij(?)/((?)xj)),Tβγαis the Christoffel syrnbol of the second kind of(N,h)(ⅲ)ΔσΨis on the normal bundle of N,whereΨ=iοΦ,i is the isometrical imbedding; (ⅳ)ΔσΨ+trgA(Φ)(dΦ,dΦ)=0([34]), (10) where A(Φ) is the second fundamental form of N in Rk.Let W1,2(M,Rk) be the Sobolev space of the maps from M to Rk whose squares of weak derivation is integral with Hilbert norm Define W1,2(M, N):={Φ∈W1,2(M, Rk):Φ(x)∈N a.e. x∈M}. We callΦ∈W1,2(M, N) a weakly harmonic map if it is a critical point of energy functional, that isΦsatisfies the Euler-Lagrange equation (10) in the sense of distributions ([34]). An energy minimizing map (E—minimizing map)Φ∈W1,2(M,N) means E(Φ)< E(Ψ) holds for any mapΨ∈W1,2(M, N) withΦ=Ψon the boundary of M. It is easy to check that an E—minimizing mapΦ∈W1,2(M, N) is a weakly harmonic map. A point x∈M is a regular point ifΦis continuous in a neighborhood of x. Let (?)= (?)(Φ) be the set of all regular points and (?)=(?)(Φ) be the complement of (?) in M, i.e.(?) is the set of all singular points. Let Rj be a j—dimensional Euclidean space. We callΦa tangent map if it is a harmonic map from Rj\{0} to any Riemannian manifold which is constant on the rays starting from the origin. If the tangent map is energy minimizing on any compact subset of Rj, it is called the minimizing tangent map. ([42])The existence of energy minimizing maps can be obtained by the direct method of the calculus of variations. The regularities of energy minimizing map between Riemannian manifolds have been studied in [42] and [43]. Using their methods, we consider the case that the source manifolds are Finsler and get the following two regularity theorems which is a generalization of the results in [42] and [43].Theorem 0.6 Let (M, F) be a compact n—dimensional Finsler manifold (n≤3), and N be a compact Riemannian manifold. LetΦ:M→N be an energy minimizing map in W1,2(M, N). Then dim(?)((?)∩intM)≤n-3, where dim(?)A is the Hausdorff dimension of a set A and 6 is the singular set ofΦ, intM denotes the interior of M. If n= 3, then (?) is a discrete set of points. Moreover, if there is an integer l≥3 such that every minimizing tangent map Rj→N is trivial, where 3≤j≤l, then dim(?)((?)∩intM)≤n-l-1. If n = l+1, then (?) is a discrete set of points, and if n< l+1, (?)= 0.Theorem 0.7 Let (M, F) be a compact n-dimensional Finsler manifold (n≥3), and N be a compact Riemannian manifold. LetΦ:M→N be an energy minimizing map in W1,2(M, N). SupposeΨ∈C2,α((?)M, N) andΦ|(?)M =Ψ, where (?)M denotes the boundary of M. Then the singular set (?) ofΦis a compact subset of the interior of M. In particular,Φis C2,αin a neighborhood of (?)M.When n= 2, we get the following result. Theorem 0.8 LetΦ∈W1,2(M, N) be an energy minimizing map from Finsler surface (M, F) to a compact Riemannian manifold N. There isΨ∈C∞((?)M, N) withΦ|(?)M=Ψin case that M is with boundary (?)M. ThenΦis smooth.Remark 2. When (M,F) is a Finsler surface without boundary and N is a sphere, Theorem 0.8 can be seen as a corollary of the result in [36].0.3 Harmonic Maps and Minimal SubmanifoldsThe variation of the volume functional for submanifolds is another varia-tion problem in Finsler geometry except for the variation of energy functional for maps, and its critical point is the minimal submanifolds ([21,48]). In [21], the author studied Finsler submanifolds by the volume form induced from the projective sphere bundle and proved that an isometric immersion is harmonic if and only if it is minimal. [59] studied the conformal maps between Riemannian manifolds and proved that the conformal immersion is harmonic if and only if it is both minimal and homothetic when the dimension of the source manifold n is greater than 2; while the conformal immersion is harmonic if and only if it is minimal when n= 2.The conformal invariance of harmonic maps on Finsler surface has been showed in [36]. Here we give another simpler proof and get two existence theo-rems. Proposition 0.9 When dimM=2,there is a stable hatmonic mapΦ:(M,F)→(M,h),where F is a conformally flat Finsler metric and h is a flat Riemannian metric.Proposition 0.10 When dimM=2,there is a stable harmonicmapΦ:(M,h)→(M,F),where h is an arbitrary Riemannian metric and F is locally Minkowski.Then we investigate the relations between harmonic maps and minimal sub-manifolds when the immersion between Finsler manifolds is conformal. It is a joint work with Kang Lin.Proposition 0.11LetΦ:(M,F)→(M,F)be a homothetic immersion be-tween Finsler manifolds.ThenΦis harmonic if and only ifΦ(M)is a minimal submanifold of(M,F).Let入:=supx∈Mλ(x). (11)入is called the uniformity constant([15]).When M is Riemannian,λ=1.λ(x) is not less than the reversibility‖I‖:=supx∈M‖I‖(x),where‖I‖(x)is the norm of the Cartan formⅠon x∈M,and‖I‖is the norm ofⅠon(M,F)([50]).When M is Riemannian,‖I‖=0.Theorem 0.12LetΦ:(Mn,F)→(M,F)be a conformal immersion.If n>2,Φis harmonic and‖I‖λ<(n-2)/2,thenΦis a homothetic immersion,whereλis the uniformity constant,‖I‖is the norm of the Cartan fromⅠon(M,F).Remark 3.When n>2 andΦis a conformal immersion between Riemannian manifolds,‖I‖λ<(n-2)/2 holds.So Theorem 0.11 and Theorem 0.12 generalize the results in[59]. 0.4 H—harmonic maps M. Ara ([4]) introduced the notations of H—energy and H—harmonic maps between Riemannian manifolds (it is called F—energy and F—harmonic map in [4]) which generalized harmonic maps, P—harmonic maps and exponentially harmonic maps. There are many works on harmonic maps between Finsler man-ifolds ([33], [34], [45], [20], [22], [44]). [26] and [27] studied P-harmonic maps and exponentially maps between Finsler manifolds, respectively. In this chapter, we study the general case of the above harmonic maps. We got the first and second variation formulas and established some Liouville's theorems. Moreover, we introduced the H—stress-energy tensor and studied its properties.Note that [28] also studied the H—harmonic maps between Finsler manifolds (it is called F—harmonic map in [28]). But we gave another form of the first variation formula which concluded a brief expression of the second variation formula and simplified the proof of the Liouville's theorems. Let H:[0,∞)→[0,∞) be a C2 strictly increasing function andΦ: (Mn,F)→(Mm,F) be a non-degenerate smooth map, i.e. ker(dΦ)= 0. We define the H-energy EH(Φ) bywhere|dΦ|2= gijΦiαΦjβgαβ.It is the energy ([33], [45], [20]), the P-energy ([26]) and the exponential energy ([27]) when H(t)= t, (2t)P/2/P(P≥4) and et, re-spectively. We shall say thatΦis an H-harmonic map if it is a critical point of the H—energy functional. A H—harmonic map is said to be stable if its second variation of the H—energy functional is always nonnegative; otherwise unstable. ([28]) After a direct computation we will get the first variation formula of the H—energy functional.Theorem 0.13 LetΦ: (Mn, F)→(Mm,F) be a non-degenerate smooth map, andΦt be a smooth variation withΦ=Φ0 with the variation field V((?)Φt)/((?)t)|t=0. Then the first variation of the H—energy functional forΦis whereτH (Φ)＝▽lH (QdΦl), HenceΦisⅡ—harmonic if and only if for allβ.Remark 4.Q denotes n(n|dΦ|P-2+F2/2|dΦ|yiyjP-2,nexp{(|dΦ|2)/2}+(F2)/2gij{exp{(|dΦ|2)/2)}}yiyj resp.)whenΦis a harmonic([20])(P—harmonic([26]),exponentially harmonic ([27]),resp.)map.There is another form of the first variation formula of the H—energy func-tional in(28]: whereWe callΦαstrongly H-harmonic map if TH(Φ)=0. By using the divergence formula on the fiber of the proj ection sphere bundle,we can get(13)from(12) directly.Moreover,we obtained two examples of H—harmonic maps.Example 1:The identity map Id:(M,F)→(M,F)is H—harmonic for any Finsler manifold(M,F).Example 2:Harmonic maps with constant energy density are H—harmonic maps.In particular,whenΦis an isometrical immersion,the following statements are equivalent:(ⅰ)Φis a minimal immersion;(ⅱ)Φis harmonic；(ⅲ)Φis H—harmonic.Using(12),we will get the second variation formula of the H—energy func-tional by a direct computation. Theorem 0.14 LetΦ:(Mn,F)→(Mm,F) be a non-degenerate smooth map, andΦt beαsmooth variation withΦ=Φ0 with the variation field V=(?)Φt/((?)t)|t=0. Then the second variation of the H—energy functional forΦisRemark 5. The second variation formula of the H—energy functional for maps between Riemannian manifolds has been given in [4]. For harmonic maps, P—harmonic maps and exponentially harmonic maps, the second variation for-mulas of the energy functional have been got in [20], [26] and [27], respectively.When (M,F) is Riemannian, we have the following theorem.Theorem 0.15 LetΦbe an H—harmonic map from a Finsler manifold (M, F) to a Riemannian manifold (M,F) with nonpositive sectional curvature. If H"≥0; thenΦis unstable.Remark 6. Theorem 0.15 has been given in case that M is Riemannian ([4]). WhenΦis harmonic, it has been got in [20]. P-harmonic maps and exponentially harmonic maps satisfy the condition H″≥0.By an extrinsic average variational method in the calculus of variations ([53]) we got three Liouville theorems.Theorem 0.16 LetΦbe a non-degenerate H—harmonic map from a unit spere Sn to a compact Finsler manifold (Mm, F). If thenΦis unstable. Remark 7. For harmonic maps and P-harmonic maps, (14) means n>2 ([20]) and n> P([26]), respectively. When|dΦ|2< n-2, (14) holds for an exponentially harmonic mapΦ([27]).Theorem 0.17 Let (Mm,g) be a submanifold of the Euclidean space Rm+q with flat normal bundle. Let Ha= (hαβα)m×m for a=m+1,…,m+q, where h is the second fundamental form of Mm in Rm+q. If Ha is positive definite everywhere for everyαand the principal curvatures satisfy whereλmaxα= max{λ1α,…,λmα},λminα= min{λ1α,…,λmα}, then there is no nonconstant stable H-harmonic map from any Finsler manifold M to such Rie-mannian manifold Mm.Remark 8. [59] studied the stability of harmonic maps from Riemannian man-ifolds to hypersurfaces and case (ii) of Theorem 0.17 generalized its result. For P-Harmonic maps and exponentially harmonic maps, the principal curvatures must satisfy respectively. More-over, we found some ellipsoids as the examples of the submanifolds described in Theorem 0.17.Example 3:Let Mm= an ellipsoid in Rm+1, where 0<α1≤…≤αm+1. Letμ:= there is no nonconstant stable harmonic (P-harmonic, exponentially harmonic, resp.) maps from any Finsler manifold M to such ellipsoid Mm. [60] studied the stability between Riemannian manifolds and ellipsoids, so Theorem 0.17 generalized its results. Theorem 0.18 LetΦbe a nonconstant H-harmonic map from a compact Finsler manifold (Mn,F) to a unit spere Sm. If thenΦis unstable.Remark 9. For harmonic maps and P-harmonic maps, (15) means m>2 ([45]) and m> P ([26]), respectively. When｜dΦ｜2< m-2, (15) holds for an exponentially harmonic mapΦ([27]).[4] studied the H-stress-energy tensor for maps between Riemannian man-ifolds. [34] studied the stress-energy tensor for maps from Finsler manifolds to Riemannian manifolds. [22] studied the stress-energy tensor for maps between Finsler manifolds. We introduce the H-stress-energy tensor defined by for non-degenerate smooth maps between Finsler manifolds and generalized the results in the above references.Theorem 0.19 LetΦ:(M,F)→(M,F) be a non-degenerate smooth map between Finsler manifolds, then the following equation holds for the H-stress-energy tensor SH(Φ): (divgSH(Φ))(l)=-< TH(Φ),dΦl>g, where divg denotes the divergence on (SM,g). Particularly, whenΦis strongly H-harmonic, (divgSH(Φ))(l) = 0.SH(Φ) is called horizontally divergence-free if∑in=1(DeiSH(Φ))(ei,Y)=0 for all Y∈C(HTM), where D is the Levi-Civita connection of the Riemannian manifold (SM,g),{ei} is any orthonormal basis for the horizontal space HTM.Theorem 0.20 LetΦbe a submersion from a Finsler manifold (M, F) to a Riemannian manifold (N,h). Then any two of the followings implies the third one:(ⅰ)Φis strongly H-harmonic; (ⅱ)the H-stress-energy tensor SH(Φ)is horizontally divergence-free;(ⅲ)the canonical projection mapπ:SM→M has minimal fibers.Theorem 0.21 LeΦ:(M,F)→(M,F)be a non-degenerate smooth map between Finsler manifolds,Ψbe a smooth section on the pulled-back cotangent bundleπ*T*M with compact support.Then the following equation holds for the H-stress-energy tensor SH(Φ): where divg denotes the divergence on(SM,g),c▽H denotes the horizontal co-variant differential with respect to the Chern connection.Theorem 0.22 LetΦ:(M,F)→(M,F)be a confomally strogly H-harmonic map between Finsler manifolds,and where n=dim M.ThenΦmust be homothetic.Remark 10.(16)means b≠2([22]),b≠P and |dΦ|2≠n-2 for harmonic maps,P-harmonic maps and exponentially harmonic maps,respectively.